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Learning objectives: Construct an appropriate null hypothesis and alternative hypothesis and distinguish between the two. Differentiate between a one-sided and a two-sided test and identify when to use each test. Explain the difference between Type I and Type II errors and how these relate to the size and power of a test. Understand how a hypothesis test and a confidence interval are related.


20.14.1. Peter Parker at Betalab Bank emailed a survey to the bank's customers. The survey included a question that asked them to rank their customer satisfaction on a scale from one to 10. She received 51 responses, and she considers that a random sample (n = 51). Among this sample, the average satisfaction score (on a scale of one to 10) is 8.50 with a sample standard deviation of 1.90. Betalab's CEO is Mary-Jane, FRM, and she hopes that the bank's average customer satisfaction is at least 9.0. Mary-Jane holds the FRM designation so she understands that acceptance of the null is more accurately a failure to reject the null, but she is a practical person. Her null hypothesis is that the population's average customer satisfaction is at least 9.0 (i.e., H0: μ ≥ 9.0 and H1: μ < 9.0). Peter shares his sample findings with five of his colleagues, and each colleague gives different input, as follows:

I. Albert says the test statistic is (8.5 - 9.0) ÷ [1.90 / SQRT(51)] = -1.88, or |-1.88| = 1.88​
II. Betty says that if the sample size were doubled, ceteris paribus (i.e., same sample mean and sample standard deviation), the test statistic will increase about +41%​
III. Chris says that (for n = 51) Mary should use a one-sided test, and with one-sided 95.0% confidence (aka, 5.0% significance) she should reject the null​
IV. Derek says that (for n = 51) Mary can accept (aka, fail to reject) the CEO's null hypothesis with 95.0% confidence but only if she artificially switches to a two-sided hypothesis (i.e., H0: μ = 9.0 and H1: μ ≠ 9.0)​
V. Erin agrees with Chris and says that Mary should use a one-sided test per the CEO's one-sided hypothesis but notes that Peter can accept (aka, fail to reject) the null with one-sided 99.0% confidence (aka, 1.0% significance)​

Which of the five statements is (are) correct?

a. None of the statements are correct
b. Only I. and II. are correct
c. Only IV and V. are correct
d. All five of the statements are correct

20.14.2. The Fulcrum Jetpack is a high-risk, high-reward leveraged exchange-traded note (ETN). It claims a (population) mean monthly excess return of at least 200 basis points. Over the last five years (i.e., sample size, n = 60 months), the sample excess mean return was 230 basis points with a standard deviation of 120 basis points. Each of the following is true EXCEPT which is false?

a. If the p-value is 0.02880 then the power of the test is 97.120%
b. In the case of the one-sided null hypothesis test, we should reject at 95.0% confidence but we can accept (aka, fail to reject) at 99.0% confidence
c. If we increase confidence from 95.0% to 99.0% (aka, decrease the significance from 5.0% to 1.0%), ceteris paribus, the power will decrease
d. We can increase the power either by decreasing the confidence level (aka, increasing the significance level) and/or, ceteris paribus, increasing the sample size

20.14.3. Janice has been asked to backtest her firm's 95.0% one-day value at risk (VaR) model. If she assumes exceptions (i.e., days when the loss exceeds the VaR level) are independent, then the binomial distribution describes the number of exceptions. The historical sample is 1,000 days based on 250 days per year. The firm's 95.0% confident one-day VaR is $38,000. If the VaR model is accurate, she expects to observe 5.0% * 1,000 days = 50 exceptions, but that is just the average of a binomial distribution. She also knows that if n*p and n*(1-p) are greater than 10, she can approximate the binomial with the normal (as a rule of thumb); indeed, 5%*1,000 > 10. Using the normal approximation, her 95.0% confidence interval is given by 1,000*5.0% +/- [1.96 × SQRT(5% × 95% × 1,000)], or (36.5, 63.5). In addition, each of the following statements is true EXCEPT which is false?

a. If she increases the confidence level of the hypothesis test (on the same 95.0% VaR model) from 95.0% to 99.0%, the power of her test will decrease
b. If the standard deviation of the VaR is approximately $4,000, then the confidence interval around her VaR is $38,000 +/- 1.96 × $4,000/SQRT(1,000) or ($37,752; $38,248).
c. She can increase the power of her hypothesis test by decreasing its confidence level (e.g., from 95% to 90.0%) but this will increase the probability of a Type I error
d. If she increases the confidence level of the hypothesis test (on the same 95.0% VaR model) from 95.0% to 99.0%, the 99.0% confident interval is given by (32.2, 67.8) exceptions

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